410Hw05ans - STAT 410 Fall 2009 Homework #5 (due Friday,...

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STAT 410 Fall 2009 Homework #5 (due Friday, October 2, by 3:00 p.m.) 1. Let X and Y have the joint probability density function f X, Y ( x , y ) = < < < + otherwise 0 1 0 4 x y y x Let U = X Y and V = X. Find the joint probability density function of ( U, V ), f U, V ( u , v ). Sketch the support of ( U, V ). X = V, Y = V U . 0 < y 0 < u , y < x u < v 2 , x < 1 v < 1. J = 1 1 0 2 v u v - = v 1 - . | J | = v 1 . f U, V ( u , v ) = f X, Y ( v , v u ) | J | = v v u v 1 4 + = 2 4 1 v u + , 0 < v < 1, 0 < u < v 2 , f U, V ( u , v ) = 0 otherwise.
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2. Let X and Y have the joint probability density function f X, Y ( x , y ) = < < < + otherwise 0 1 0 4 x y y x a) Find f Y | X ( y | x ). f X ( x ) = ( 29 + x dy y x 0 4 = ( 29 0 2 2 x y y x + = 3 x 2 , 0 < x < 1. f Y | X ( y | x ) = 2 3 4 x y x + , 0 < y < x , 0 < x < 1. f Y | X ( y | x ) is undefined for x < 0 or x > 1. b) Find E ( Y | X ). E ( Y | X = x ) = + x dy x y x y 0 2 3 4 = 0 3 2 2 3 4 2 3 1 x y y x x + = 18 11 x , 0 < x < 1. E ( Y | X = x ) is undefined for x < 0 or x > 1. E ( Y | X ) = 18 X 11 . c) Find f X | Y ( x | y ). f Y ( y ) = ( 29 + 1 4 y dx y x = y y x x 1 2 4 2 + = 2 2 9 4 2 1 y y - + , 0 < y < 1. f X | Y ( x | y ) = 2 2 9 4 2 1 4 y y y x - + + , y < x < 1, 0 < y < 1. f X | Y ( x | y ) is undefined for y < 0 or y > 1.
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d) Find E ( X | Y ). E ( X | Y = y ) = - + + 1 2 2 9 4 2 1 4 y dx y y y x x = y y x x y y 1 2 3 2 2 4 3 2 9 4 2 1 1 + - + = 2 3 2 9 4 2 1 3 7 2 3 1 y y y y - + - + = y y y 2 9 2 1 3 7 3 7 3 1 2 + + + = ( 29 ( 29 9 1 3 7 7 1 2 y y y + + + = y y y 27 3 14 14 2 2 + + + , 0 < y < 1. E ( X | Y = y ) is undefined for y < 0 or y > 1. E ( X | Y ) = Y 27 3 Y 14 Y 14 2 2 + + + .
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3. Once a car accident is reported to an insurance company, the company makes an initial estimate, X, of the amount it will pay to the claimant. When the claim is finally settled, the company pays an amount, Y, to the claimant. The company has determined that X and Y have the joint p.d.f. f ( x , y ) = ( 29 ( 29 ( 29 1 1 2 2 1 2 - - - - x x y x x , x > 1, y > 1. a) Given that the initial claim estimated by the company is 1.5, determine the probability that the final settlement amount exceeds 2. f X ( x ) = ( 29 ( 29 ( 29 - - - - 1 1 1 2 2 1 2 dy y x x x x = 3 2 x , x > 1. f Y | X ( y | x ) = ( 29 ( 29 1 1 2 1 - - - - x x y x x , y > 1. f Y | X ( y | x = 1.5 ) = 4 3 - y , y > 1. P ( Y > 2 | X = 1.5 ) = - 2 4 3 dy y = 8 1 = 0.125 . b) Find E ( Y | X = x ). E ( Y | X = x ) = ( 29 ( 29 - - - - 1 1 1 2 1 dy y x x y x x = x , x > 1.
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4. 2.2.4 Let X 1 and X 2 have the joint pdf h X 1 , X 2 ( x 1 , x 2 ) = 8 x 1 x 2 , 0 < x 1 < x 2 < 1, zero elsewhere. Find the joint pdf of Y 1 = X 1 / X 2 and Y 2 = X 2 . Hint:
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This note was uploaded on 10/14/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Fall '08 term at University of Illinois at Urbana–Champaign.

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410Hw05ans - STAT 410 Fall 2009 Homework #5 (due Friday,...

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